Geometric singular perturbation theory, mainly due to Tikhonov and Fenichel [18–21], is an efficient tool for investigating the slow-fast dynamics of the systems with two timescales. According to the geometric singular perturbation theory, the dynamical behavior of the two timescale systems is governed by the structure of the slow manifold including the shape, stability, and bifurcation of the slow manifold, as well as the location and stability of the equilibrium of the two timescale systems.
First, the geometric singular perturbation theory defines the slow manifold of equation (2.6) as the equilibriums of the fast subsystem equation (2.7)
$$\begin{aligned} M =& \biggl\{ (Z,Y,V,U)\Bigm| Y F_{0} \biggl( \frac{Z}{Y},Y; \alpha^{2} \biggr)=0 \biggr\} \\ =& \biggl\{ (Z,Y,V,U)\Bigm| Y= \frac{X}{(\alpha^{2}+X^{2})(1-X)} \biggr\} , \end{aligned}$$
(3.1)
where \(X=\frac{Z}{Y}\). It is the phase space for the reduced problem. Denote \(\Psi(X; \alpha^{2}):=\frac{X}{(X^{2}+\alpha ^{2})(1-X)}\), \(\Pi(X; Y_{\max},\varrho):=Y_{\max}(1-\varrho X)\). From \(Y= \frac {X}{(\alpha^{2}+X^{2})(1-X)}\), we have
$$\begin{aligned} \frac{dY}{dX}=\Psi' \bigl(X; \alpha^{2} \bigr)= \frac{\mathbb {P}_{3}(X)}{(X^{2}+\alpha^{2})^{2}(1-X)^{2}}, \end{aligned}$$
(3.2)
where \(\mathbb{P}_{3}(X)=2X^{3}-X^{2}+\alpha^{2}\).
The shape of the slow manifold depends on the coefficient \(\alpha ^{2}\). In the following discussion, the case with an S-shape curve of the slow manifold is focused on, correspondingly, we let \(\alpha^{2}\) satisfy \(0<\alpha^{2} < \frac{1}{27}\) (see Figure 1).
To determine the stability and bifurcation of the slow manifold M, let us consider the stability of the equilibriums of the fast subsystem equation (2.7). For any equilibrium point \((Y_{0},X_{0})\), the linearized system is
$$\begin{aligned} \frac{dZ}{dt}= \biggl[1-2X_{0}-\frac{2\alpha^{2} X_{0}}{Y_{0}(\alpha^{2} +X_{0}^{2})^{2}} \biggr]Z(t). \end{aligned}$$
(3.3)
In the slow manifold M, the one eigenvalue of equation (3.3) is
$$\begin{aligned} \lambda=1-2X_{0}-\frac{2\alpha^{2} X_{0}}{Y_{0}(\alpha^{2} +X_{0}^{2})^{2}}. \end{aligned}$$
As for λ, there are two critical points \(X_{1}\) and \(X_{2}\) satisfy with \(0< X_{1}<\frac{1}{3}<X_{2}\leq\frac{1}{2}<1\) such that \(\lambda>0\) with \(X_{0}\in(X_{1},X_{2})\) and \(\lambda<0\) with \(X_{0}\in(0,X_{1})\cup(X_{2},1)\). The stable branches and the unstable branch meet in bifurcation points which can be shown to represent two saddle-node points [1], denoted by \(B_{1}=(Z_{1},Y_{1},V_{1},U_{1})\) and \(B_{2}=(Z_{2},Y_{2},V_{2},U_{2})\), where \(Z_{i}=X_{i}Y_{i}\), \(V_{i}=U_{i}=Y_{i}\), \(\frac{X_{i}}{(X_{i}^{2}+\alpha^{2})(1-X_{i})}=Y_{i}\), and \(\mathbb {P}_{3}(X_{i})=0\) (\(i=1,2\)). The slow manifold M is divided into three parts by the bifurcation points \(B_{1}\) and \(B_{2}\),
$$\begin{aligned} M=M_{1}+M_{2}+M_{3}, \end{aligned}$$
here
$$\begin{aligned}& M_{1}= \biggl\{ (Z,Y,V,U)\Bigm|Z=XY, Y=\frac{X}{(X^{2}+\alpha^{2})(1-X)}, 0< X< X_{1} \biggr\} , \\& M_{2}= \biggl\{ (Z,Y,V,U)\Bigm|Z=XY, Y=\frac{X}{(X^{2}+\alpha^{2})(1-X)}, X_{1}< X< X_{2} \biggr\} , \\& M_{3}= \biggl\{ (Z,Y,V,U)\Bigm|Z=XY, Y=\frac{X}{(X^{2}+\alpha^{2})(1-X)}, X_{2}< X< 1 \biggr\} , \end{aligned}$$
and \(M_{2} \) is unstable, \(M_{1}\) and \(M_{3}\) are stable, as illustrated in Figure 2.
Next, consider the location and stability of the equilibrium points of equation (2.6), the equilibrium points are the intersecting nodes of the slow manifold and the curved surface,
$$\begin{aligned} \bigl\{ (Z,Y,V,U)|Y=Y_{\max}(1-\varrho X) \bigr\} . \end{aligned}$$
The number and location of the equilibrium points are dependent on the parameters of the system; there may be one, two or three equilibrium points, as shown in Figure 1 and Figure 3 with the projection on the \((X,Y)\) plane.
To decide on the stability of those equilibrium points, consider the characteristic equation
$$\begin{aligned} D(\lambda)=\lambda^{4}+b \lambda^{3}+c \lambda^{2}+d\lambda+e=0, \end{aligned}$$
(3.4)
where
$$\begin{aligned}& b=\varepsilon(2a-\varrho X_{0} ), \\& c=\varepsilon \bigl(a^{2}+\varrho X_{0}-\varrho X_{0}^{2} \bigr)-\varepsilon^{2} \cdot2a\varrho X_{0}, \\& d=\varepsilon^{2} \biggl[a\varrho X_{0}(2-2 X_{0}-a)+\frac {a^{2}Y_{0}}{Y_{\max}} \biggr], \\& e=\varepsilon^{2}a^{2}\varrho X_{0}(1-X_{0}). \end{aligned}$$
When \(\varepsilon=0\), \(D(\lambda)=0\) has four characteristic roots
$$\begin{aligned} \lambda_{0}^{i}=0,\quad i=1,2,3,4. \end{aligned}$$
When \(0<\varepsilon\ll1\), the three characteristic roots of \(D(\lambda )=0\) can be described as
$$\begin{aligned} \lambda_{\varepsilon}^{1}=\varepsilon\kappa_{1},\qquad \lambda_{\varepsilon }^{2}=\varepsilon\kappa_{2},\qquad \lambda_{\varepsilon}^{3}=\varepsilon\kappa_{3},\qquad \lambda _{\varepsilon}^{4}=\varepsilon\kappa_{4}, \end{aligned}$$
where \(\kappa_{i}=O(1)\) (\(i=1,2,3,4\)) are constant. The equilibrium point \((Z_{0},Y_{0},V_{0},U_{0})\) is stable if and only if all the four characteristic roots \(\lambda_{\varepsilon}^{i}\) (\(i=1,2,3,4\)) are with a negative real part. Since e and c are always positive, the sign of \(\lambda_{\varepsilon }^{i}\) (\(i=1,2,3,4\)) is one of \((-,-,-,-)\), \((-,-,+,+)\), and \((+,+,+,+)\).
When \(\varrho<\frac{2a}{X_{0}}\) and \(\varrho X_{0}(2-2 X_{0}-a)+\frac{a Y_{0}}{Y_{\max}}>0\), one has \(-b<0\) and \(-d<0\), which leads to \(\lambda_{\varepsilon}^{i}<0 \) (\(i=1,2,3,4\)) due to \(\lambda_{\varepsilon}^{1}+\lambda_{\varepsilon}^{2} +\lambda_{\varepsilon}^{3}+\lambda_{\varepsilon}^{4}=-b<0\) and \(\lambda_{\varepsilon}^{1}\lambda_{\varepsilon}^{2}(\lambda _{\varepsilon}^{3}+\lambda_{\varepsilon}^{4}) +\lambda_{\varepsilon}^{3}\lambda_{\varepsilon}^{4}(\lambda _{\varepsilon}^{1}+\lambda_{\varepsilon}^{2})=-d<0\).
When \(\varrho>\frac{2a}{X_{0}}\) or \(\varrho X_{0}(2-2 X_{0}-a)+\frac{a Y_{0}}{Y_{\max}}<0\), one has \(-b>0\) or \(-d>0\), which leads to the sign of \(\lambda _{\varepsilon}^{i} \) (\(i=1,2,3,4\)) being either \((-,-,+,+)\) or \((+,+,+,+)\).
Thus, when the equilibrium point \((Z_{0},Y_{0},V_{0},U_{0})\in M_{2}\) and \(\varrho>\frac{2a}{X_{0}}\) or \(\varrho X_{0}(2-2 X_{0}-a)+\frac{a Y_{0}}{Y_{\max}}<0\), then the equilibrium point located in \(M_{2}\) is unstable. When the equilibrium point \((Z_{0},Y_{0},V_{0},U_{0})\in M_{1}\cup M_{3}\), \(\varrho<\frac{2a}{X_{0}}\), and \(\varrho X_{0}(2-2 X_{0}-a)+\frac{a Y_{0}}{Y_{\max}}>0\), then the equilibrium point located in \(M_{1}\cup M_{3}\) is stable.
Once the structure of the slow manifold M and the position and stability of the equilibriums of equation (2.6) are obtained, the dynamical behaviors of equation (2.6) can be analyzed through the geometric singular perturbation theory.
Conditions \(\varrho<\frac{1}{X_{2}}\) and \(-Y_{\max}\varrho>\frac {Y_{2}-Y_{1}}{X_{2}-X_{1}}\) guarantee that there is an equilibrium point in the slow manifold \(M_{1}\) and \(M_{3}\), respectively, as shown in Figure 3(b). When \(\frac{1}{X_{2}}<\varrho<\frac{1}{X_{1}}\) and \(-Y_{\max}\varrho <-\frac{Y_{1}}{X_{2}-X_{1}}\), there are no equilibrium points in the slow manifold \(M_{1}\cup M_{3}\), as shown in Figure 1. According to the geometric singular perturbation theory, the solution trajectory will be attracted by the stable manifold and repelled by the unstable manifold.
Remark 2
The original model (1.1) in [1] is a 2D system. The Jacobian J of the vector field defining the original system is given by second-order matrices. The authors used \(\operatorname{tr}(J)\) and \(\operatorname{det}(J)\) to judge on the positive and negative of eigenvalues. However, the corresponding characteristic equation of our model is a quartic equation. Thus, the positive and negative judgment of eigenvalues is more complicated. Through the analysis of the characteristic equation, the positive and negative of characteristic roots were obtained.
When the solution trajectory is attracted by the stable slow manifold \(M_{1}\), it will stick to the stable manifold and move slowly from down to up, due to
$$\frac{dY}{d\tau}=Y f \biggl(\frac{Z}{Y},V;\varrho,Y_{\max} \biggr)>0 $$
with \((Z,Y,V,U)\in M_{1}\). When the solution trajectory crosses the saddle-note bifurcation point \(B_{1}\), it will be repelled by the unstable slow manifold \(M_{2}\), and be attracted by the stable slow manifold \(M_{3}\) quickly, then the solution trajectory will stick to \(M_{3}\) and move slowly from up to down due to \(\frac{dY}{d\tau}<0\) with \((Z,Y,V,U)\in M_{3}\). When the solution trajectory crosses the saddle-note bifurcation point \(B_{2}\), it will be again attracted by \(M_{1}\) quickly. Thus, the slow movement process alternates with the rapid movement process in the predator-prey system, and the predator-prey system undergoes relaxation oscillation, as shown in Figure 4.
Remark 3
Reference [1] used the attraction domain of the upper and lower stable branch of the quasi-steady state (16) to judge on the clockwise direction of the relaxation oscillation. In this paper, we use the sign of \(\frac{dY}{d\tau}\) to judge on the clockwise direction of the relaxation oscillation. This method is simpler and clearer in comparison.
The relaxation oscillation of (1.2) is described approximately as
$$ \textstyle\begin{cases} Y=\frac{X}{(X^{2}+\alpha^{2})(1-X)},& X\in (X_{2}^{*},X_{1}],\\ Z=Y_{1}X_{1},& X\in (X_{1},X_{1}^{*}],\\ Y=\frac{X}{(X^{2}+\alpha^{2})(1-X)},& X\in[X_{2},X_{1}^{*} ),\\ Z=Y_{2}X_{2},& X\in [X_{2}^{*},X_{2} ), \end{cases} $$
(3.5)
where \(\frac{X_{i}^{*}}{((X_{i}^{*})^{2}+\alpha^{2})(1-X_{i}^{*})}=Y_{i}^{*}\).
Since \(0<\varepsilon\ll1\), the quick movement process of fast manifolds is instantaneous, thus, the period of the relaxation oscillation is governed by the slow movement process. Hence, from (3.2) the asymptotic expression for the period T of the relaxation oscillations to the lowest order in the perturbation parameter ε is [22–24]
$$\begin{aligned} T =& \int_{M_{1}}\,d\tau+ \int_{M_{3}}\,d\tau+ O(\varepsilon) \\ \approx& \int_{M_{1}}\frac{dY}{Yf(X,V;\varrho,Y_{\max})}+ \int _{M_{3}}\frac{dY}{Yf(X,V;\varrho,Y_{\max})} \\ =& \int_{X_{2}^{*}}^{X_{1}}\frac{\Psi'(X)\,dX}{\Psi(X)f(X,\Psi (X);\varrho,Y_{\max})} + \int_{X_{2}}^{X_{1}^{*}}\frac{\Psi'(X)\,dX}{\Psi(X)f(X,\Psi(X);\varrho ,Y_{\max})} \\ =& Y_{\max} \biggl( \int_{X_{2}^{*}}^{X_{1}}\frac{\mathbb{P}_{3}(X)\,dX}{X\mathbb{P}_{4}(X)} + \int_{X_{1}^{*}}^{X_{2}}\frac{\mathbb{P}_{3}(X)\,dX}{X\mathbb {P}_{4}(X)} \biggr), \end{aligned}$$
(3.6)
where \(O(\varepsilon)\) is the so-called junction time governed by the fast movement process, and \(\mathbb{P}_{4}(X)=Y_{\max}\varrho X^{4}-Y_{\max}(1+\varrho)X^{3} + Y_{\max}(1+\alpha^{2} \varrho)X^{2} -(1+\alpha^{2} Y_{\max} +\alpha^{2}\varrho Y_{\max})X +\alpha^{2} Y_{\max}\).
In summary, the main results of this paper can be stated as follows.
Proposition
If one of the following two conditions holds:
- (H1):
-
\(\max\{\frac{1}{X_{2}},\frac{2a}{X_{2}},\frac {2a}{X_{1}}\}<\varrho<\frac{1}{X_{1}}\), \(-Y_{\max}\varrho<-\frac{Y_{1}}{X_{2}-X_{1}}\);
- (H2):
-
\(\frac{1}{X_{2}}<\varrho<\frac{1}{X_{1}}\), \(-Y_{\max}\varrho<-\frac{Y_{1}}{X_{2}-X_{1}}\), and
\(\varrho X_{i}(2-2 X_{i}-a)+\frac{a Y_{i}}{Y_{\max}}<0\) (\(i=1,2\)),
then the predator-prey system undergoes relaxation oscillation, and the analytical expressions of the relaxation oscillation and its period are described approximatively as equations (3.5) and (3.6), respectively.
Remark 4
Conditions (H1) and (H2) are incompatible.